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Abstract

One of the important open questions in the theory of free--surface ideal fluid flows is
the dynamic stability of traveling wave solutions. In a spectral stability analysis, the
first variation of the governing Euler equations is required which raises both theoretical
and numerical issues. With Zakharov and Craig and Sulem's formulation of the Euler
equations in mind, this paper addresses the question of analyticity properties of first
(and higher) variations of the Dirichlet--Neumann operator. This analysis will have
consequences not only for theoretical investigations, but also for numerical simulations
of spectral stability of traveling water waves.